Frequency response analysis of continuous flow systems

Frequency

response

analysis

of continuous

flow systems

H. KICAMEBSand G. AIBERDA Laboratorium voor Phyi ,che Technologie, Tectmologioal University, Delft, Netherlands (Received a Ju?lc1958) Summary-The distribution of residence times in a continuous flowsystem can de deducedfrom experiments concerning the bebaviour of longitudirml concentration gradlenta on their course through the system. In this paper the application of sinusoidallyv8rylng concentrationa is treated from 8 theoretical and experimental viewpoint. As an illuntr8tion of this frequency response analysis, experimental resulta are given for longltudirml diffusion in liquid flow through packed Raschii rings and for back-mixing of 8 liquid flowing over the packing of an absorption column. R&W&--L8 repartition des “ dun&esde &jour ” de l’&oulement dans les syst&mes en continu peut Atre deduit de la fawn, dont un gradient longltudinal de k concentration se prop8ge a travers le aysteme. Ici on traite l’application dea variations slnusoidale~ de la concentration, du point de vue theOrlque et experimentale. On pr6sente dea rt%ultats exp&iment8ux de c&e armlyse harmonique: la diffusionlongitudhmle dans un liquide qui traverse un llt flxe d’anneaux Raschig et le melange longitudinal d’un liquide descendant dana une colonne. d’absorption B

1.

mixing in the system, the response curve to a step-wise disturbance is an exponentid &suction. In between these two extreme types of F-&grams one can find nearly all the response curves observed in practice. Their shape can be attributed to a number of phenomena for which various expressions are in use : longitudinal or axial diffusion, back-diffusion, back-mixing, shortcircuiting, by-passing, channelling, trapping in dead corners, non-perfect mixing.

INTRODUCTION

The main purpose of this paper is to direct attention towards the possibilities of frequency response analysis of continuous processes in We will mainly deal chemical engineering. here with only one espect of a continuous process, viz. the distribution of residence times in continuous flow systems. In this respect this paper can be regarded as a counterpart of a recent publication by DANCKWERTS [l], to which we shall frequently refer. DANCKWERTShas given a general discussion on the distribution of residence times, which can be found by investigating the mode of propagation of an initial disturbance through the system. The distribution effect is best visualized if one suddenly changes a property (e.g. concentration) of the incoming stream from one steady value to another. The value of this property in the outflow as a function of time gives the response curve of the system to this Such a response initial step-wise disturbance. curve corresponds with the F-diagram in DANCKWERTS’ paper. If the residence time is equal for all elements of the fluid entering the system, the response curve is again exactly a step function which is delayed with respect to the original disturbance by the time of residence. On the other hand if there is complete

The aim of introducing a step-wise disturbance at the inlet of the continuous flow system is to obtain information regarding the actual distribution of residence. times. Generally speaking, the same information can be obtained by analysing the response curve resulting from any kind of initial disturbance. As special ca3es the step-wise disturbance and also the delta-function (instantaneous injection) are convenient because their relation to the distribution function of the residence times is apparent. On the other hand they have two disadvantages. First, it is difficult to produce a sharp step - or deltafunction in the concentration of a flowing liquid or gas exactly at the entrance of the system under investigation. With increasing demand for accurate results and with decreasing variation in residence times, the introduction of a signal of sufficient discontinuity in a stream becomes

178

El.KRAMERS und G. ALBERDA : Prequencyresponseanalysis of continuousflow systems.

impossible. In the second place, one encounters rate coefficients of heat or mass transfer in mathematical complexities when trying to physical operations. Very recently, WILHELM calculate the response curves on the basis of an and DEISLER [‘?I, [3] have reported the first assumed physical mechanism. These are inherent results of frequency response analysis applied in the non-stationary behaviour of the system to the inter- and intra-particle diffusion in gas under investigation. flow through a fixed bed of porous catalyst carrier. The above difficulties can be partly overcome In the present paper we shall restrict ourselves by applying a sinusoidal disturbance to the to the purely hydrodynamical problem of the incoming stream. The measurement of the distribution of residence times. Firstly a brief amplitude attenuation and the phase shift survey will be given of the theoretical solutions between the signals at the outlet and the entrance, for a few simple types of flow systems. Secondly as a function of the frequency of the signal, some experimental viewpoints will be discussed give complete information on the distribution together with illustrative results on longitudinal of residence times of a system. The frequency diffusion in liquid flow through Raschig rings response diagram thus obtained can eventually and on the back-mixing of liquid flowing down be transformed into a distribution curve by one a packed absorption column. of the practical methods of Fourier synthesis [2]. In the following we shall denote the deviation The main advantages of this harmonic analysis of the sinusoidal signal (e.g. tracer concentration) are that no discontinuities have to be introduced from its average value by c. For purposes of into the fluid streams, and that a steady cycling calculations it is customary to regard the signal condition can be obtained. Furthermore the as a complex harmonic function with a circular stationary operation simplifies the mathematical frequency w. We then have for the value of the treatment of a few special problems, such as the signal at any place in the system : case of a number of mixers in cascade or that c = X eiwr. of piston flow with longitudinal diffusion. Appli(1) cation of theory is restricted to so-called linear X is a complex magnitude which can be systems, where the dynamic behaviour is inderepresented by a radius vector in the complex pendent of the amplitude of the entering signal. plane, having a certain length (modulus) and As long as this signal does not interfere with the phase angle (argument) with respect to the flow phenomena in the system this condition is positive real axis. In the frequency response satisfied. From the experimental point of view analysis the relationship between two signals c the frequency response method requires more or their vectors X is of interest. It can be elaborate equipment than the measurement of a expressed by the ratio of their respective moduli response to a step- or delta-function. (a) and the difference (+) between their respective The characterization of the dynamic behaviour arguments. If we take for the two signals the of linear systems by the frequency response concentrations of the effluent (c,) of the system method is very well known in the field of and of the incoming stream (ci), the complex servomechanisms and automatic control. The ratio between X, and Xi is called the harmonic dynamic behaviour of a complete process involvresponse function of the entire system. This ing, e.g. chemical reaction, transfer of heat or function depends on the frequency w of the mass transfer can be investigated by the same applied signal in a way wnich is characteristic technique with a view to the possible applications of the system. It contains both the amplitude of automatic control. PROFOS[3] and TAKAHASHI ratio (IXe/Xil = CC)and the phase shift + of X, [4] have published such experiments on heat with respect to Xi and may be expressed as exchangers. A more general review of these follows : problems has been presented by RUTHERFORD[S] and by AIKMAN[6]. Frequency response measureX -4 = c&4. (2) ments may also give information regarding the Xi 174

H. KRAMERSand G. ALBERDA:

Frequencyresponseanalysisof continuousflow systems.

In flow systems the outgoing signal is always smaller in amplitude and retarded in phase with respect to the entering signal, so generally a < 1 and + < 0. A graphical representation of a and + as a function of the frequency w is called a frequency response diagram. Such a diagram can be obtained by measuring the amplitude ratio and the phase shift between an outgoing and incoming sinusoidal signal ; for a number of special cases the harmonic response function can be calculated in a rather simple manner. 2.

PERFECT

Fig. 1 gives the frequency response diagrams for systems containing different numbers of

MIXERS

By definition the concentration in a perfect mixer has a value at every point equal to the concentration at the outlet c,. For continuous flow through the mixer we have the material balance :

where ? is’the average time of residence : volume of the mixer divided by the constant volumetric flow rate. If cj is varied sinusoidally, we fined for the steady state after substitution of (1) : xi=xe+iw7x,,

(4)

and for the harmonic response function of one perfect mixer : X 2

1

=

Fig. 1. Frequency response diagram for n equal mixers

Cjzz+

Xi

.n cascade (----

) and for pkton flow with longitudinal diffusion (- - -).

The values of a and 4 are found to be : a1 = (1 + 3 &

=

-

;i2)-*

tan-l W? . >

(6)

Applying the result (5) to a system containing n perfect mixers in cascade having equal times of residence aln, we get for the response function between X, and Xi :

$gl+iwy with the following values of a and + :

nixers in series with a total average residence time ;i. A special case of (7) presents itself for n --f OJ : Lim(1

+i~$+=e-i+,

(9)

n+co so that here : a,

= 1 (10)

4,

(7)

= -cu.?

This case is represented by perfect piston flow where no amplitude attenuation of the incoming 175

l-l. KMBXERS and G. fbBEEDA : Frequency reqnmse analysis of continuous Bow systems.

signal occurs and where the phase lag is causec by the pure distance-velocity lag. There is nc mixing or diffusion in the direction of flow. The frequency response diagram of a rea system generally lies in between those for one perfect mixer and for perfect piston flow However, it is not at all necessary that tht diagram for a real system coincides with one fol a certain number of perfect mixers. It is to be noted the equations (5), (7) and (9: have the same form as those for the fractiona: completion of a first order chemical reaction ir continuous flow through well agitated vessels, The relation between the reactant concentration leaving and entering one perfect mixer is in thal case represented by eq. (4) if the operator a’~ is replaced by the first order reaction velocity constant. Chemical reactions in continuous stirred tank systems have been amply discussed by MCMULLIN and WEBER (O), ELDRIDGE and PIBET (10) and SCHOENEMANN(11). The harmonic response function of any system of stirred tanks, with or withbut external recirculation can be calculated with the same relative ease as the yield of a first order chemical reaction in these systems. The extraction of the amplitude and phase relationships from the complex response function presents no difficulties, although in some cases it may involve elaborate computational work. 8.

PISTON FLOW WITHLONGITUDINAL DIFFUSION

In a number of systems the distribution of residence times can be regarded as the combined result of perfect piston flow and a coefficient of longitudinal diffusion D. Generally the role of molecular diffusion is rather unimportant. In flow through fixed beds of solids and in beds in which par&late fluidization occurs the smallscale fluctuations of the intergranular fluid velocity cause a statistical distribution of residence times which can be interpreted as the result of an ” eddy ” diffusivity. The same applies to a tubular reactor in which a uniform degree of turbulence is created by mechanical means. The introduction of a longitudinal diffusivity has significance only if the underlying phenomenon is of a statistical nature, i.e. if the 176

elementary fluctuations are repeated many times during the course of the fluid through the system. Marcrophenomena, such as by-passing and trapping, make the picture of diffusion inadequate. The differential equation for the concentration in a system (Fig. 2) with uniform flow velocity D=o

D-O ---___

D+o

--___

_

--CB

Ci

0,

---__

i

*=o

Fig. 2.

-0

--_-_

V*L

Packed tube with piston flow nnd longitudmal diffusion.

u and a longitudinal diffusivity D is :

3% +-Day

ii bc -----=

13c

D3t

o

.

(11)

It is now assumed that the system has a finite length L and that a constant diffusivity D exists over the cross-sectional area and for D < y < L. For y < 0 and 2/ > L, D = 0. This leads to the following boundary conditions :

uc,=~-D?,y== dY

snd

3C

*a =~ic-_D-,y=L.dY

(12) (18)

l?he latter condition has to be supplemented by the condition cc = c,., which, appears to be necessitated by the physical conditions at the ?xit. Therefore (18) has to be replaced by : 3C _=o, 3Y

y=L.

[n order to solve eq. (11) for a sinusoidally rarying input signal ci and for steady conditions ue introduce the substitution given by eq. (1). IJhe partial differential equation (11) then educes to the following ordinary linear differen:ial equation with constant coefficients :

dzX w-sdy

v dX ---

i”x,o, D

(15)

I-I. KMMEES

and C. ALBEBDA

: Frequency respomeanalyds oi continuous flow system.

with the boundary conditions :

x,=x-fdX,y=O l.4 dY -= dx 0 *Y- -L. dY

1

(16)

These equations are identical with those given by DANCKWERTS(El], eq. (80), (81) and (82) ) for the reactant concentration resulting from a first order chemical reaction in a similar flow system. The harmonic response function X/X, can be deduced diitly from DANCKWERTS’ general solution (eq. (88)) if the reaction velocity constant k is replaced by iw. For the reponse function of the entire system we have : g= kp (1 +ze

=Pgv

4-u

-m=P$l

+i (17)

where p = 4-e This function depends on two variables for which we choose the dimensionless groups The latter number U&L/G= wv and 2DyuL. can be regarded as the ratio between the average time of residence L/ii = F and the diffusion time constant for the length L : 70 = La/2D. For F/To = tD/ilL > 1 , d COnCCnhtiOn differences are levelled out relatively rapidly by diffusion. For the limiting of D = co, we f3nd that eq. (17) is indentical with the response function for one perfect mixer, eq. (5). As F/T~ decreases, the influence of diffusion becomes less important, In the limiting case F/To = 2D/EL = 0 eq. (17) yields the solution for perfect piston flow, eq. (9). For small values of 2D/GL, eq. (17) can be approximated by

Comparison of (19) and (20) shows that perfect piston flow with longitudinal diffusion yields the same frequency response diagram as a number of n perfect mixers with the same total residence time Ti, if

The physical interpretation of this equation is as follows. Suppose we divide the reactor of Fii. 2 into n parts of equal length L/n. For this length the average time of residence is L/nii and the diffusion time constant is Ls/2Dns. Apparently, when the relation (21) holds, both times are equal. The validity of equation (21) is restricted to very high values of n or of U/2D. The diagram of Fig. 1 shows the amplitude curves for various values of EL/SD. Comparing these with the curves for perfect mixers one is inclined to accept the more general connection between n and D : 1 1 - GL n-

-2.

w

This analogy may be used for n greater than 5 to 10, depending on the accuracy required. It also holds for n = 1, but is a quite insufficient approximation in the intermediate region. The phase lags of two systems which are analogous in the sense of eq. (22) are not materially different for WV< 5, mainly because in this region the phase lag is rather insensitive to changes in n or D/CL. The practical advantage of the analogy (22) is that for flow systems have a relatively small diffusivity the frequency response diagrams can be calculated from eq. (8) which is more convenient in handling than eq. (17). 4.

POISEUILLX FLOW

The harmonic response function for laminar Bow in a tube with a circular cross-section is By personal communication, we were informed that Dutch/Shell Laboratory, Amsterdam) arrived at the -me analogy by comparing the step-fin&ion response functiolIf# of the two type8 of system. l

Da. G. H. REMAN(Royal At the same time eq. (5) for n equal mixers may be approximated for large values of n by : I77

,) : ’

H.

FRAMERS

andC.

ALBERUA:

Frequency response nnnlybiisof continuous flow systems.

illustrative of cases where the llow rate is nonuniform. In the following we assume that there is neither radial nor axial diffusion and that the velocity distribution is given by

(28)

The frequency response diagram calculated from (24) has been drawn in Fig. 8. It shows less phase lag and more amplitude damping than piston flow. Its phase behaviour might be compared with that of 8 perfect mixers in series, at least for UT < 5. The amplitude attenuation however is much less than for 8 mixers. This feature is indicative of a non-uniform velocity distribution which also may be caused by channelling and trapping. 5.

Fig.

a. Frequency response diagram for Poiseuilleflow

in a tube (L

) & comparea with that of s equal mixers in series ().

The time of residence for a streamline at a distance r from the axis is u/L. As there is no backmixing in the streamline its response function represents a pure distance-velocity lag, exp -iwL/u according to (9). Integrating for all streamlines multiplied by their velocities one finds for the harmonic response function between the average outlet signal and the incoming signal :

REMARKS ON THE EXPERIMENTAL TECHNIQUE

For frequency response measurements on a continuous flow system one needs first of all a good property to use as a “ signal ” and a means of measuring it. In the second place a mechanism for introducing the signal sinusoidally is required. For measuring the distribution of residence times the concentration of any material which is miscible with the main flow and does not disappear by reaction, adsorption or desorption, can be used as a “ signal.” Considerable variations in concentration may cause undesirable convection currents as a result of densitygradients. The choice of the tracer component is further restricted by the requirements of the concentration measurement : a quick response with respect to the lags of the system, continuous measurement or recording, a high degree of accuracy and linearity. For liquid flow the most convenient “ signal ” is the concentration of an appropriate electrolyte in water, which can be measured by the electric conductivity. In the case of gas flow, currently available devices for measuring gas composition have a response which is too slow to be used for this kind of investigation. Only the gas analyser described by DEISLER and WILHELM [8], [12] seems to meet the special requirements in all respects. With this instrument one measures the ionization current in a gas which is exposed to alpha-radiation from a radio-active source. It indicates concentrations linearly and has a time lag of the order of one second. Although for liquids and gases radioactive tracers could be used, this method appears to be very costly if continuous indication or recording of the concentration is required. There are many possible devices for producing

178

a constant flow with a sinusoidally varying Generally it amounts to the concentration. addition of a concentrated solution or mixture to the main stream, at a relatively small rate which is varied periodically with different frequencies. This flow can be given the desired fluctuation by simply turning an appropiate valve, by mechanically varying the discharge of a feeder pump or by using a constant discharge pump combined with a sinusoidally varying buffer volume. A more costly way of regulating the additional flow incorporates an automatic controller the setpoint of which is varied in a sinusoidal way. For quantitative measurements it is important that the entering signal contains no higher harmonics. The latter are damped relatively quickly so that too small an amplitude ratio a may be found. In the experiments to be reported in the next section, we used a variable stroke pump for feeding the concentrated solution. The signal obtained after mixing with the main stream contained higher harmonics due to the reciprocating action of the pump and to some backlash in the mechanism for varying the stroke,. These harmonics were filtered off in a packed column of suitable length in the same way as has been reported by DEISLER and WILHELM [8]. 6.

accurate one, measurements

we have that for

deduced from these both flow velocities

EXPERIMENTAL RESULTS

The results of some preliminary measurements will be discussed here in order to illustrate the preceeding sections and to show the feasibility and the limitations of the frequency response technique. Longitudinal mixing was investigated for flow of water through a tube (diameter 7.4 cm., length 84 cm.) with 10 mm Raschig rings (fractional free volume 0.7~). Fig. $ shows the resulting frequency response diagram of X,/Xi for the two water velocities investigated (l-2 and 2.2 cm/set). In the amplitude diagram the experimental points lie in between the curves for sL/2D = 15 and 20. The points in the phase diagram suggest a smaller value of the order of 10, the discrepancy probably being caused by a certain amount of trapping in the interior of the rings. As the amplitude curve is the more 179

Fig. 4. Frequency response measurements of liquid low through 8 packed tube (Raschig rings lonun).

Relating D to the nominal ring diameter we have

D ig

= g

x 4

4

= 1-l f

d,,

iid7

0.1 ; loo < T<200?

l!he observed longitudinal diffusivities are about 10 times greater than those reported by WILHELM 171for radial diffusion with liquid flow through a led of packed spheres. As with radial diffusion, L proportionality between D and ;il seems to :xist in the flow region covered. [l] found from the response curve 0 8 step-wise disturbance of 8 similar system with about he same packing and u = O-4cm/see : Dfiir = l-8.

t DANCKWEEITS

II. KMMIW

and G. ALRBRDA : Frequency

response onalysie of

Frequency response measurements for water flow in the same tube (diameter 7.4 cm, lengtl~ 84 cm and u = 2-l cm/set) without any packing yielded the results which have been plotted in Fig. 8. Although under these conditions laminar flow would be realized in a sufficiently long tube, here the turbulence created by the sudden enlargement at the liquid inlet causes the system to behave in about the same manner as three equal mixers in series, which is purely accidental.

continuous flow systuns.

mixed with the shortest possible lag before its concentration was recorded. The average residence time, which depends on the liquid hold-up of the system, was determined from the measured phase shifts for small OF values. In this region 4 should be equal to - ti in view of the rather small amount of back-mixing. Experimental runs were made at three liquid flow rates +&, both without and with air flow & which was adjusted somewhat below the flooding rate. The experimental results have been plotted in Fig. 5 for the amplitude diagram only, together with the theoretical cuwes for Z.L/2D = 10 and 20. Evidently the results cannot be ascribed to the presence of longitudinal diffusivity only. Presumably this is due to the non-uniform velocity distribution which is well known to occur in packed absorbers as a consequence of preferential wetting near the wall. On the other hand the results clearly show that back mixing of the liquid is increased with decreasing liquid load and increasing gas load. The latter effect was, of course, to be expected, but it appears that under normal operating conditions it would not seriously impair the separating efficiency of an absorber. ACKNOWLEDGEMENT

The first assistance

author wishes to acknowledge the of Messrs. .I. H. WERVERS and TAN BUN SENG who performed a part of the experiments mentioned.

Fii. 6. Amplitude-frequency characteristics of liquid flow over the packing of an ahsorption column with and without counter-curTent gas flow.

Finally, we mention a number of experiments on the distribution of residence times of water running over the packing in an absorption column with a countercurrent air flew. These measurements were carried out in a 15 cm diameter column with a 66 cm high packing of 10 mm Raschig rings, The water feed was uniformly distributed over the top of the packing. The outgoing liquid was separated from the gas and

REFERENCES [l] DANCKWEETS, P. V.; Chem. Eng. Sci. 1958 2 1. [Z] H~xos, L. VON, J~Nssow, B., PenssoN, Th.; Acta Polytechnica 1952 112, Physica end Applied Mathematics Series vol. 2, nr. 8. [S] Profoe, P.; Vektorieile Regeltheorie, Dips. Zurich, 1248. [4] TAKA~ASE~, Y.; Regelung3to&nik 1958 1 82. [S] RUTEERW~D, C. I.; Proc. Inut. Mech. Engr3. (London) 1958 162 884. [6] AIK~AN, A. R.; Regehmg3technik 1958 1 4. [7] WILE~~~, R. II.; Chem. Eng. Progress 1888 49 158. [s] DBIBLER, P. F., WILEELM, R. II.; Ind. Eng. Chem. 1958 4S 1219. [S] MCMULLIN, R. B., WEBEB, M.; Trans. Am. Inst. Chem. Engm. 1985 31 488. [lo] ELDBIWE, J. W., PIBET. E. L.; Chem. Eng. prosresa 1958 46 228. [ll] SCEOENEMANN. K.; De&ems hionographs 1952 21 208. [If] D~sLE~, P. F.; Dirs. Princeton University, 1952, Par1 II.

180

I-I. KuMEIw

and G. hBERDA

: hequeucy

response an&&

of continuous flow systems.

NOTATION c = deviation of concentration from average value c, entering the system ; c, leaving

;

the system

4 = nomiaal diameter of Rawhig rlnga D = apparent longitudinal dBusivity L = length of tube or column n=numberofequalmixeminaeriea r = variable distance irom tube axis R=radhU oftube t==thne u = ilow velocity ; 5 average (intqgruwk) velocity X = symbol representing concentration vector in the complex plane g~kngthcQordiMteinthedlnctionofnow a = amplitude ratio between outgoing and bwoming eoncentxation c v=-ticviacoaity 7=averagetimeofresidenceinthesystem + -phMe shift between outgoing and incoming concentition c (radians) Q) = circuhu f=¶uency (~/=c)

IS1